Rigorous solutions, Berry phases and quantum-classical correspondence for time-dependent non-Hermitian SU(2)-SU(1, 1) Hamiltonians

Abstract

Based on the generalized gauge transformation and invariant methods, we study in this work the rigorous solution for non-Hermitian Hamiltonians consisting of SU(1, 1) and SU(2) generators, which describe the harmonic oscillator and a single spin of arbitrary spin value, respectively, in periodically time-dependent fields. A non-Hermitian Hamiltonian with real eigenvalues is called pseudo-Hermitian. Such a Hamiltonian can be converted to a Hermitian one by a similarity transformation. We extend the concept of pseudo-Hermitian Hamiltonians to the time-dependent case. Under the generalized gauge transformation with a time-dependent Hermitian operator, the Hamiltonian becomes Hermitian and time-independent in the new gauge. The rigorous solution, as well as the Berry phase, is obtained by applying the inverse transformation to return to the original gauge. The time-dependent Schrödinger equation is also solved in terms of non-Hermitian invariant operators, which possess conserved real eigenvalues. We derive the time-dependent solutions and corresponding geometric phases by eigenstate expansion of the invariant operators, combined with the time-evolution operator. It is observed that the non-Hermitian coupling constant indeed has an exceptional point Gc(ω) for the SU(2) non-Hermitian Hamiltonian, where all eigenstates are degenerate with zero eigenvalue and the Berry phase diverges. The eigenvalues become complex beyond Gc(ω) . However, the eigenvalues are real within the entire region of coupling constant for the SU(1, 1) system, different from the common belief that the exceptional point must exist in the non-Hermitian Hamiltonian. The classical counterpart of the SU(1, 1) non-Hermitian Hamiltonian is, in fact, a complex function of position-momentum variables. It can be converted to a real function via the generalized gauge transformation, accompanied by a canonical-variable transformation. This process corresponds exactly to the Hermitization procedure of the pseudo-Hermitian Hamiltonian. We obtain the classical Hannay angle for the periodically time-dependent SU(1, 1) pseudo-Hermitian Hamiltonian. The non-adiabatic Hannay angle and the Berry phase satisfy precisely the quantum-classical correspondence, γn(T)=−(n+12)∆Θ .